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Fluid limit theorems for stochastic hybrid systems with application to neuron models

Part of: Limit theorems Markov processes Physiological, cellular and medical topics

Published online by Cambridge University Press:  01 July 2016

K. Pakdaman ,
M. Thieullen  and
G. Wainrib
K. Pakdaman*
Affiliation:
Université Paris VII
M. Thieullen*
Affiliation:
Université Paris VI
G. Wainrib*
Affiliation:
Ecole Polytechnique, Université Paris VII and Université Paris VI
*
∗ Postal address: Institut Jacques Monod UMR7592, Bâtiment Buffon, 15 rue Hélène Brion, 75205 Paris cedex 13, France.
∗∗ Postal address: Laboratoire de Probabilités et Modèles Aléatoires UMR7599, Boîte 188, 175 rue du Chevaleret, 75013 Paris, France.
∗∗∗ Email address: wainrib.gilles@ijm.univ-paris-diderot.fr
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Abstract

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In this paper we establish limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamics coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump Markov processes. We prove a functional law of large numbers with exponential convergence speed, derive a diffusion approximation, and establish a functional central limit theorem. We apply these results to neuron models with stochastic ion channels, as the number of channels goes to infinity, estimating the convergence to the deterministic model. In terms of neural coding, we apply our central limit theorems to numerically estimate the impact of channel noise both on frequency and spike timing coding.

Keywords

Stochastic hybrid systempiecewise-deterministic Markov processfluid limitLangevin approximationkinetic modelneuron modelHodgkin-HuxleyMorris-Lecarstochastic ion channels

MSC classification

Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles 60J75: Jump processes
Secondary: 92C20: Neural biology 92C45: Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 3 , September 2010 , pp. 761 - 794
Copyright
Copyright © Applied Probability Trust 2010 

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